Membrane bending is critical for assessing the thermodynamic stability of proteins in the membrane

The ability of biological membranes to bend is critical to understanding the interaction between proteins and the lipid bilayer. Experimental and computational studies have shown that the membrane can bend to expose charged and polar residues to the lipid headgroups and water, greatly reducing the cost of protein insertion. However, current computational approaches are poorly equipped to accurately model such deformation; atomistic simulations often do not reach the time-scale necessary to observe large-scale rearrangement, and continuum approaches assume a flat, rigid bilayer. In this thesis we present an efficient computational model of a deformable membrane for probing these interactions with elasticity theory and continuum electrostatics. To validate the model, we first investigate the insertion of three membrane proteins and three aqueous proteins. The model finds the membrane proteins and aqueous proteins stable and unstable in the membrane, respectively. We also investigate the sensitivity of these predictions to changes in several key parameters. The model is then applied to interactions between the membrane and the voltage sensor segments of voltage-gated potassium channels. Despite their high numbers of basic residues, experiments have shown that voltage sensors can be stably accommodated in the membrane. For simple continuum electrostatics approaches that assume a flat membrane, the penalty of inserting these charged residues would seem to prohibit voltage sensor insertion. However, in our method the membrane deforms to enable interaction between solvent and the charged residues. Our calculations predict that the highly charged S4 helices of several potassium channels are in fact stable in the membrane, in accord with experimental observations. Experimental and computational evidence has shown that the cost for inserting multiple charged amino acids into the membrane is not additive; it is not as costly to insert a second charge once a first has already been inserted. Our model reflects this phenomenon and provides a simple mechanical explanation linked to membrane deformation. We additionally consider the energetics of passive ion penetration into the membrane from bulk solvent. We use coarse-grained molecular dynamics to guide our input parameters and show that ion permeation energy profiles agree with atomistic simulations when membrane bending is included

Continuum approaches to understanding ion and peptide interactions with the membrane.

Experimental and computational studies have shown that cellular membranes deform to stabilize the inclusion of transmembrane (TM) proteins harboring charge. Recent analysis suggests that membrane bending helps to expose charged and polar residues to the aqueous environment and polar head groups. We previously used elasticity theory to identify membrane distortions that minimize the insertion of charged TM peptides into the membrane. Here, we extend our work by showing that it also provides a novel, computationally efficient method for exploring the energetics of ion and small peptide penetration into membranes. First, we show that the continuum method accurately reproduces energy profiles and membrane shapes generated from molecular simulations of bare ion permeation at a fraction of the computational cost. Next, we demonstrate that the dependence of the ion insertion energy on the membrane thickness arises primarily from the elastic properties of the membrane. Moreover, the continuum model readily provides a free energy decomposition into components not easily determined from molecular dynamics. Finally, we show that the energetics of membrane deformation strongly depend on membrane patch size both for ions and peptides. This dependence is particularly strong for peptides based on simulations of a known amphipathic, membrane binding peptide from the human pathogen Toxoplasma gondii. In total, we address shortcomings and advantages that arise from using a variety of computational methods in distinct biological contexts

APBSmem: A Graphical Interface for Electrostatic Calculations at the Membrane

Electrostatic forces are one of the primary determinants of molecular interactions. They help guide the folding of proteins, increase the binding of one protein to another and facilitate protein-DNA and protein-ligand binding. A popular method for computing the electrostatic properties of biological systems is to numerically solve the Poisson-Boltzmann (PB) equation, and there are several easy-to-use software packages available that solve the PB equation for soluble proteins. Here we present a freely available program, called APBSmem, for carrying out these calculations in the presence of a membrane. The Adaptive Poisson-Boltzmann Solver (APBS) is used as a back-end for solving the PB equation, and a Java-based graphical user interface (GUI) coordinates a set of routines that introduce the influence of the membrane, determine its placement relative to the protein, and set the membrane potential. The software Jmol is embedded in the GUI to visualize the protein inserted in the membrane before the calculation and the electrostatic potential after completing the computation. We expect that the ease with which the GUI allows one to carry out these calculations will make this software a useful resource for experimenters and computational researchers alike. Three examples of membrane protein electrostatic calculations are carried out to illustrate how to use APBSmem and to highlight the different quantities of interest that can be calculated

The Journal of Membrane Biology Continuum Approaches to Understanding Ion and Peptide Interactions with the Membrane Continuum Approaches to Understanding Ion and Peptide Interactions with the Membrane

Abstract Experimental and computational studies have shown that cellular membranes deform to stabilize the inclusion of transmembrane (TM) proteins harboring charge. Recent analysis suggests that membrane bending helps to expose charged and polar residues to the aqueous environment and polar head groups. We previously used elasticity theory to identify membrane distortions that minimize the insertion of charged TM peptides into the membrane. Here, we extend our work by showing that it also provides a novel, computationally efficient method for exploring the energetics of ion and small peptide penetration into membranes. First, we show that the continuum method accurately reproduces energy profiles and membrane shapes generated from molecular simulations of bare ion permeation at a fraction of the computational cost. Next, we demonstrate that the dependence of the ion insertion energy on the membrane thickness arises primarily from the elastic properties of the membrane. Moreover, the continuum model readily provides a free energy decomposition into components not easily determined from molecular dynamics. Finally, we show that the energetics of membrane deformation strongly depend on membrane patch size both for ions and peptides. This dependence is particularly strong for peptides based on simulations of a known amphipathic, membrane binding peptide from the human pathogen Toxoplasma gondii. In total, we address shortcomings and advantages that arise from using a variety of computational methods in distinct biological contexts

Ion solvation free energy (kcal/mol).

<p>Ion solvation free energy (kcal/mol).</p

States used to compute ion solvation energies.

<p>(A) KcsA ion channel (orange) embedded in a slab of low-dielectric material (gray) with two ions in the selectivity filter (blue) and a single ion in the water filled cavity (red). For clarity only two subunits are shown. (B) Set up in panel A without the cavity ion. (C) The cavity ion in bulk water in the absence of KcsA and the membrane. The ion solvation energy is calculated by computing the total electrostatic energy of each system in A, B and C and then calculating the quantity: .</p

Convergence properties of test cases I–III.

<p>We computed the absolute value of the percent error, , for each test case using a number of different discretization values. All energies are reported with respect to the solution value at the finest level of discretization, , which was 0.312 Å in test Case I and 0.375 Å in Cases II and III. Values along the x-axis are spaced using a log base 2 scale. In all graphs, the number of grid points used to achieve the grid spacing on the x-axis was 17, 33, 65, 97, 129, and 161 (). (A) Convergence of the protein solvation energy, Case I. A grid spacing of 0.512 Å gives a solution 1.5% of the highest resolution value. The energy values smoothly converge as the resolution increases. (B) Convergence of the ion solvation energy, Case II. The error monotonically decreases as the level of discretization increases. At  = 0.625 Å the energy value is within 2.5% of the final value. (C) Convergence of the gating charge energy in the closed state, Case III. Rather than report the gating charge, here we plot the energy of the closed state. This method converges much more quickly than the other Cases since it does not involve Born Self energy terms. The energy at the second finest level is 0.33% of the value at the finest level. Even at a grid spacing of  = 0.938 Å the computed energy is within 3% of the best value. In all cases, the convergence properties and the accuracy of the solutions depend critically on the refinement of the protein surface boundaries. Here we use the spl2 method for charge mapping in APBS, which gives very desirable convergence properties.</p

States used to compute protein solvation energies.

<p>(A) The helix (orange) is pictured embedded in the membrane, which is delineated by the upper blue and lower gray lines. The membrane core between the two red lines is assigned a dielectric value  = 2. A headgroup region of 8 Å is indicated between the water and membrane core. Bulk water above and below the membrane is assigned a dielectric value of  = 80. (B) The helix in the bulk water ( = 80) in the absence of the membrane. The helix carries one charged residue (Arg14) shown in green in (A) and (B). The protein solvation energy is calculated by computing the total electrostatic energy of systems A and B and then calculating the quantity: . Images rendered with VMD <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0012722#pone.0012722-Humphrey1" target="_blank">[63]</a>.</p